Boolean Parenthesization problem

Given a boolean expression, a string with True or False as operands and between each pair of operand,  there is boolean operator (and &, or | and xor ^). Find number of ways in which this Boolean expression can be parenthesized so that expression evaluates to True. This is known as Boolean Parenthesization problem. To understand problem better, let’s take some examples
Expression :

T ^ F & T

Two ways :

((T ^ F) & T) and (T ^ (F & T))

T | T & F ^ T

Four ways :

((T|T)&(F^T)), (T|(T&(F^T))), (((T|T)&F)^T)
and (T|((T&F)^T))

Boolean Parenthesization problem : Line of thoughts

What will be the most trivial Boolean expression? Of course, an expression with only one Boolean value T or Boolean value F.

How many ways can this expression be parenthesized so that expression evaluates to True ? Apparently, there is only one way.

For Boolean value T, there is one way, (T); whereas for F, there no way we can parenthesize to evaluates True. An expression can evaluate to either True or False value.

Let’s say, T(i,j) is number of ways expression from i to j can be parenthesized so that it evaluates to True. Similarly, F(i,j) is number of ways expression evaluates to False. With base case, only one value either T or F is there, hence i=j, hence following equations hold true

```T(i,i) = 1 if operand is T
0 if operand is F

F(i,i) = 0 if operand is T
1 if operand is F```

How to calculate T(i, j) for expression with more than one values and operators between them?  This is something familiar to matrix chain multiplication problem. We will put parenthesis at all possible position and count how many ways these two resultant expressions hold True. Once we have count for each expression, we can combine count based on operator between split expression.

For expression from index i to index j, find k such that i<k<j, and find number of ways expressions from i to k and k+1 to j evaluates to True. Interesting, once these numbers are determined, number of ways for expression i to j can be calculated based on operator between expression i to k and k+1 to j.

When Boolean operator is & (AND)

When can expression (i,j) be True if expression is of form Expression(i, k) & Expression(k+1, j)?  Only if Expression(i,k) and Expression(k+1,j) are  both True. Hence, for any k, expression can be True in T(i,k) * T(k+1, j) where T(i,k) is number of ways Expression(i,k) is True and T(k+1, j) is number of ways Expression(j+1, j) is True. For all possible values of k, expression becomes

`T(i,j)  = Summation ( T(i,k) * T(k+1,j)) for all k such that i < k < j`

How about expression (i,j) being evaluates to False? Simple enough, one of the two expression should evaluate to False.

If Total(i,j) represents total number of ways an expression can be parenthesized irrespective of out being True or False, then

```Total(i,j) =  Total(i,k) * Total(k+1, j)
or
Total(i,j) = T(i,j) + F(i,j)```

If we take out number of ways an expression can parenthesized as True from Total, it gives number of ways it can be evaluates False. Hence, below equation

```F(i,j) = Sum ( Total (i,j) - T(i,k)* T(k+1)) for all k for i< k< j
or
F(i,j) = Sum (Total(i,k) * Total(k+1, j) - T(i,k)* T(k+1) )```

When Boolean operator | (OR)

In case, operator is OR, then, whole expression is True is any one of the expressions is True. How many ways both Exp(i,k) and Exp(k+1, j) be False.

Following the same logic from AND operator True, it can be derived that

`F(i,j) = Summation (F(i,k)* F(k+1,j)) for all  i<k<j`

Overall expression is True when both sub-expressions are not False. Hence.

`T(i,j) = sum ( Total(i,j) - F(i,k)* F(k+1,j)) for k such i<k`

In the same vein, T(i,j) and F(i,j) when operand is xor will be

`T(i,j) = sum(T(i,k)*F(k+1,j) + F(i,k)* T(k+1,j)) for k such i<k`

To find solution to Boolean parenthesis problem, find is T(1,N).

Implementation : Boolean parenthesization problem

```package com.company;

/**
* Created by sangar on 31.12.17.
*/
public class BooleanParenthesis {

public static int calculateNumberOfWays(String operators, String operands){
int numOperands = operands.length();

int[][] F = new int[numOperands][numOperands];
int[][] T = new int [numOperands][numOperands];

for (int i=0; i<numOperands; i++){
System.out.println(operands.charAt(i));
F[i][i] = (operands.charAt(i) == 'F')? 1: 0;
T[i][i] = (operands.charAt(i) == 'T')? 1: 0;
System.out.println(T[i][i]);
}

for (int L=1; L<numOperands; L++) {
for (int i=0; i<numOperands-L; ++i){
int j = i+L;
T[i][j] = F[i][j] = 0;
for (int k=i; k<j; k++){
int totalIK = T[i][k] + F[i][k];
int totalKJ = T[k+1][j] + F[k+1][j];
if (operators.charAt(k) == '&') {
T[i][j] += T[i][k]*T[k+1][j];
F[i][j] += (totalIK *totalKJ - T[i][k]*T[k+1][j]);
}
if (operators.charAt(k) == '|'){
F[i][j] += F[i][k]*F[k+1][j];
T[i][j] += (totalIK*totalKJ - F[i][k]*F[k+1][j]);
}
if (operators.charAt(k) == '^'){
T[i][j] += F[i][k]*T[k+1][j] + T[i][k]*F[k+1][j];
F[i][j] += T[i][k]*T[k+1][j] + F[i][k]*F[k+1][j];
}
}
}
}
for(int i=0; i<numOperands; i++){
for(int j=0; j<numOperands; j++){
System.out.println("(" + i + "," + j + ") :"  + T[i][j]);
}
}
return T[0][numOperands-1];
}

public static void main(String[] args) {

String operands = "TTFT";
String operators = "|&^";

System.out.println("Number of ways to parenthisize expression : " +
calculateNumberOfWays(operators, operands));

}
}
```

Complexity of  dynamic programming approach to find ways to parenthesize a Boolean expression to evaluate it to True is O(n3). and space complexity is O(n2) .

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